There are a variety of musical tone generator systems in which the musical waveshape is first created in digital form and then converted to an analog signal for a sound system by means of a digital-to-analog converter. Representative digital tone generators of this type have been disclosed and described in U.S. Pat. No. 3,515,792, Digital Organ; U.S. Pat. No. 3,809,789, Computer Organ; and U.S. Pat. No. 4,085,644, Polyphonic Tone Synthesizer.
The simplest means, and therefore the most commonly used means, for convererting a sequence of digital numeric data into a corresponding analog waveshape is to repetitively convert the digital numbers to an analog voltage using a digital-to-analog converter. A sample and hold circuit is usually employed after such conversions so that the present current, or voltage, level is maintained at a substantially constant value until the succeeding conversion time. Such a sample and hold circuit is frquently referred to as a zero-order sample and hold. Sometimes it is also called a "box car" detector.
It is an inherent characteristic of a digital-to-analog system, wherein data is converted at periodic intervals, that the resulting analog signal has spectral components which are imaged at all integer multiples of the periodic interval of the signal conversion. This periodic interval is called the sampling period and is the period of the digital-to-analog conversion.
It is well known that a zero-order sample and hold conversion system will produce an output signal spectrum which is clustered about multiples of the sampling period and that the original input spectrum (assuming that the sampling period is sufficiently high so that there is no fold-over or aliasing) will be multiplied by a spectral amplitude factor of the form EQU G(f)=T sin(.pi.fT)/.pi.fT (Equation 1)
where T is the sampling period. A discussion of this well-known property can be found on page 135 of the book:
Cooper, G. R. and Clare D. McGillen, Methods of Signal and System Analysis. New York, Holt, Rinehart and Winston, Inc., 1967.
The effectiveness of a zero-order sample and hold circuit as an interpolation means for a digital-to-analog conversion depends upon the relative value of the sampling period T in comparison to the highest frequency component in the spectral content of the input digital data sequence. As a general rule, the higher the sampling frequency f.sub.s =1/T in comparison with the highest frequency component in the digital sequence, the better will be the suppression of undesired, or noise components in the signal output spectrum.
The term noise is used herein in a genereric sense to encompass components of the waveshape that are undesired. For example, if one of the referenced digital tone generators is intended to produce a specified tone color having 16 harmonics, then any additional harmonics produced by the digital-to-analog conversion system are considered to be noise. It is evident that such noise, which may consist of an extra harmonic, in some situations may not produce a disagreeable or objectionable sound. However, in many situations the "extra" harmonics can be very objectionable and can produce overtones which are too far removed in frequency from the pitch of the desired tone to be considered acceptable even if they may not be characterized as being disagreeable to a listener.
FIG. 1 shows a typical spectral curve for the output signal from a zero-order sample and hold circuit. The lower graph illustrates the waveshape produced by converting the sequence of digital numbers and maintaining a constant signal amplitude between the sampling times. The waveshape is synthesized from a periodic sequence having 32 equal harmonics. The upper graph is the output spectrum and exhibits the characteristic amplitude variation of the form sinx/x corresponding to Equation 1. The higher harmonic clusters only decrease very gradually with the higher frequencies.
An obvious and commonly used method to reduce the large number of undesired sampling harmonics is to employ a low pass filter following the zero-order sample and hold circuitry. The practical implementation problem is to design a low-pass filter having a sufficiently sharp cut-off so that it only attenuates the undesired frequencies without affecting the desired harmonics while still retaining a short transient time response. While low-pass filters have been used in digital-to-analog conversions systems they generally do not provide a feasible noise reduction system for musical generators of the above referenced types. For these musical generators, the cut-off frequency of the low-pass filter should be changed for each fundamental of the generated tone. Some simplification can be attained by changing the cut-off frequency only as a function of the octave in which the fundamental falls.
A noise reduction system intended for use with a digital-to-analog tone conversion system is described in the inventor's U.S. Pat. No. 4,111,090 entitled "Noise Reduction Circuit For A Digital Tone Generator." The system disclosed in the referenced patent achieves an improved attenuation characteristic for a zero-order sample and hold circuit used with a digital tone generator. The improvement does not require increasing the number of data points in each fundamental period of the tone geing generated. The improvement is obtained by providing a circuit for implementing linear interpolation between successive digital data points. In one embodiment described in U.S. Pat. No. 4,111,090, at least seven additional data points are inserted by an interpolation between each two consecutive data points of the original sequence of data points constituting the musical waveshape. The sampling rate is thereby effectively increased by a factor of eight. This is accomplished by providing a circuit arrangement in which stored data words defining the amplitudes of a succession of sample points for the waveshape are transferred successively to first and second registers at a rate determined by the fundamental pitch of the note being generated. In addition, data words as they are transferred from the first register to the second register are also transferred to the input of a digital-to-analog converter at the same predetermined rate. A subtracting and dividing means coupled to the first and second registers generates an output signal proportional to the difference in value between the digital words in the two registers. This difference signal is used to repeatedly increment the value of the input from the first register before it is applied to the digital-to-analog converter.
It is an object of the present invention to further reduce the residual noise at the output of the digital-to-analog converter to a level below that attainable with either a zero-order sample and hold circuit or such a circuit in combination with a linear interpolation system such as that described in U.S. Pat. No. 4,111,090. It is also an object of this invention to decrease the output noise without increasing the clock rates to a frequency higher than would be used to implement a system according to the referenced patent.
It is known in the signal theory art (i.e. page 138 of the above referenced book), that if a signal has frequencies f limited to a finite range of values such as -W.ltoreq.f.ltoreq.W, and if this signal is known only at discrete intervals of time t.sub.n =n/2W, -.infin.&lt;n&lt;.infin. then the original continuous signal f(t) can be completely recovered from the set of discrete samples f(n/2W) by summing weighted values of the discrete samples according to the relation ##EQU1##
Equation 2 can be written in the general form ##EQU2## where g(2Wt-n) represents the weighting function used to smooth the discrete signal amplitude values f(n/2W). Thus, at least in theory, the continuous smoothed signal function f(t) can be perfectly recovered without any extraneous sample noise if all the values of f(n/2W) are simultaneously known for all time (complete past, present, and future) and if the weighting function g(2Wt-n) is also known and is applied for all time.
If f(t) is a periodic function, which is the case for the tone generators described in the above referenced patents, then the knowledge of the sample points for a single period of the waveform is exactly equivalent to having complete knowledge of the sample points for all time. With appropriate choices for the number of sample points per waveshape period and choice of the weighting function, a finite form of Equation 2 can be employed to reconstruct a musical waveshape from an input set of discrete samples represented by a sequence of digital values.